New Dealers continue leading the country after World War II.
It might not be socialism, but it sure is social democracy lite. If Truman's Fair Deal passes or FDR or Wallace get most of the Second Bill of Rights passed, the United States of the late 1940s is in all actuality to the left of Sweden today, with guaranteed jobs and legally mandated full employment policies with universal health insurance and TVA authorities as far as the eye can see.
Basically, what I think you have to have to make this plausible is to avoid FDR's disastrous second term. Let's say that he meets with his Keynesian advisers first, they convince him not to cut spending and to continue priming the pump. And let's say that he avoids the Court Packing stuff, too. That means a lot. If the economy continues to recover and there are mass strike waves that align to bring the New Deal's policy goals to bear, the progressive majority in Congress isn't undone in 1938's Congressional elections.
That portends massive butterflies in and of itself. A stronger progressive majority in Congress means that the Republicans might end up nominating someone even to Willkie's left in 1940 and to Tom Dewey's in 1944 and 1948. So long as you can keep the momentum of reform with the progressives, stop Taft-Hartley, and keep expanding the state, you can make this work. It just requires different leadership in the postwar period.
"The American people will never knowingly adopt Socialism. But under
the name of 'liberalism' they will adopt every fragment of the Socialist
program..." - Norman Thomas